The authors give a calculation of the Hochschild homology of a simply connected topological space \(X\) with cohomology algebra of the form \[ H^* (X,K) = K[x_1, \dots, x_n]/(f_1, \dots, f_n) \tag{*} \] \(K\) a field of characteristic 0, \((f_1, \dots, f_n)\) a regular sequence such that each \(f_i\) is a polynomial only in the variables \(X_j\) of degrees less than \(\deg (f_i) - 1\). The result is obtained from an explicit calculation of the cohomology of a commutative differential graded algebra of the form \[ \Bigl( K[X_1, \dots, X_n] \otimes \Lambda (y_1, \dots, y_n) \otimes \Lambda (x_1, \dots, x _n) \otimes K [Y_1, \dots, Y_n], d \Bigr) \] with \(d(X_i) = 0\), \(d(y_j) = f_j(x_1, \dots, x_n)\) where \((f_1, \dots, f_n)\) is a regular sequence, \(d (x_i) = 0\) and \(d(Y_j) = \sum_i {\partial f_j \over \partial X_i} \otimes x_i\).
They also use this calculation to prove a conjecture of Burghelea and Vigué-Poirrier: Any simply connected space with cohomology as in (*) has quasifree cyclic homology. This conjecture has recently been proved by Vigué-Poirrier using different methods. Let \(V^*\) be a graded \(K\)-vector space with a \(K[\alpha]\)-module structure \(\nabla : K[\alpha] \otimes V^* \to V^*\), \(\deg \alpha = 2\). Then there is a \(K\)-linear map \(S : V^* \to V^{*+ 2}\) defined by \(\nabla (\alpha^p \otimes x) = S^p (x) \). Call \((V^*,S)\) free if \(S\) is injective, trivial if \(S = 0\), and quasifree if it is the direct sum of a free and trivial module.
This results are applied
\(\bullet\) to calculate the Poincaré polynomials of the Hochschild homology of \(\text{SU} (3)/T\) and \(G_2/T\) where \(T\) is the respective maximal torus
\(\bullet\) to prove that the cyclic homology of simply connected spaces with a certain minimal model is not quasifree
\(\bullet\) to prove, in particular, that Sp(20/SU(6) and \(\text{SU} (6)/ \text{SU} (3) \times \text{SU} (3)\) have nonquasifree cyclic homology.